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Chapter 2
Effect of Turbulence on Fixed-Speed Wind Generators
Hengameh Kojooyan Jafari
Additional information is available at the end of the chapter
http://dx.doi.org/10.5772/51407
1. Introduction
The influence of wind energy connection to the grid has increased greatly and turbulence or
unreliable characteristics of wind energy are expected to produce frequency and voltage
changes in power systems and protection systemequipment. To prevent these changes, it is
necessary to study the working point change due to turbulence. In other papers, the voltage
and transient stability analysis have been studied during and after turbulence [2] and the
impact of WTGs (wind turbine generators) on the system frequency, inertia response of dif
ferent wind turbine technologies, and comparison between inertia response of single-fedand doubly-fed induction generators have been examined. Moreover study of the frequency
change alone was conducted using Dig-SILENT simulator for FSWTs (fast-speed wind tur
bines) with one-mass shaft model [2].
In this chapter both frequency and grid voltage sag change are presented with MATLAB
analytically and also by SIMULINK simulation in FSWTs with one- and two-mass shaft turbine
models to compare both results and a new simulation of induction machine without limiter
and switch blocks is presented as a new work. The first part of study is frequency change
effect on wind station by SIMULINK that shows opposite direction of torque change in
comparison with previous studies with Dig-SILENT. The second part of study is effect offrequency and voltage sag change on wind station torque due to turbulence in new simula
tion of induction generator that is new idea.
2. Wind turbine model
The equation of wind turbine power is
2012 Jafari; licensee InTech. This is an open access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
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31
2 p wP ACr u= (1)
whereis air density,Ais area of turbine, Cpis power coefficient and wis wind speed.
The Cpcurve and equation are shown in Fig. 1 and given by equation (2) and (3)
97 3
8
1
15
1 2 3 4 6
9
3
8
1
1
1
pitch pitch
cc
c
p pitch pitch
pitch pitch
C c c c c c ec
c
l q qq q
l q q
- - + +
= - - - - + +
(2)
116.5 0.0021
0.44 125( 0.002) 6.94pC e l
l
- +
= + -
(3)
wherepitch is blade pitch angle, is the tip speed ratio described by equation (4). The pa
rameters are given in Table 1.
M
w
R
v
wl= (4)
where Ris blade radius.
Figure 1. Curve of Cpfor different tip speed ratios .
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The curve of Fig.1 has positive slope before Cpmaxand it has negative slope after Cpmax.
3. One-Mass Shaft Wind Station Model
Induction machine equation is
me m m
dT T J C
dt
ww- = + (5)
Where, Tmis the mechanical torque, Teis the generator torque, Cis the system drag coeffi
cient andJis the total inertia.
Table 1 shows the parameters of the one-mass shaft turbine model and induction generator.
Generator Wind Turbine
Rs= .011 c1=.44
L s= .000054H c2= 125
L m= .00287H c3= 0
L r= .000089H c4= 0
R r= .0042 [] c5= 0.1
Jm=.5 to 20.26 [kgm2] c6= 6.94
p(#pole pairs) = 2 c7= 16.5
Pn= 2e6 [w] c6= 0.1
c9= -.002
R= 35 [m]
A= R2 [m2]
=1.2041 [kg/m3]
vw= 6, 10, 13 [m/s]
pitch = 0 []
Table 1. Parameters of one- mass shaft turbine model and generator.
4. Two-Mass Shaft Induction Machine Model
This model is used to investigate the effect of the drive train or two-mass shaft, i.e., themasses of the machine and the shaft, according to the equation (8) [3], [4]. In this equation,Jtis wind wheel inertia, JGis gear box inertia and generators rotor inertia connected through
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the elastic turbine shaft with a as an angular stiffness coefficient and C as an angular
damping coefficient.
The angular shaft speed tcan be obtained from equations (6) and (7) [1], [3], [4].
TGis the torque of the machine, Ttis the turbine torque, tis the angular turbine shaft angle,
Gis the angular generator shaft angle,is the inverse of the gear box ratio and JGandJt are
the inertia of the machine shaft and turbine shaft, respectively.
The Parameters, defined above, are given in Table 2.
This model is described as equation(8).
( ) ( )GG G t GB t GBd C
T Jdt
w kd d w w
n n= - - - - (6)
( ) ( )tt t t GB t GB
dT J C
dt
wk d d w w = + - + - (7)
2 2. . .1 0
1.
0
0 0
1 0 0 0 0 00 1 0 0
G GG G G GG
t t G
t
t t t t G G t
t t
C C
J J J JJ
TC CJ
J J J J T
n n n k nk
w w
w wn nk k
w d
w d
- - -
- - - = +
&
&
(8)
1/80
JG[kg.m2] .5
J t[kg.m2] 1
C[Nm/rad2] 1e6
[Nm/rad] 6e7
Table 2. Parameters of two-mass shaft model.
5. Induction Machine and Kloss Theory
In a single-fed induction machine, the torque angular speed curve of equation (12) [1] is
nonlinear, but by using the Kloss equation (13), equations (9), (10), and (11), this curve is lin
early modified [1], [2] as shown in Fig. 2. Therefore, the effect of frequency changes in wind
power stations can be derived precisely by equation (12) and approximately using equation
(13), as shown in Figs. 26.
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( )
2
2 2
22
2 2 2
ss ss
b
sm ss rr s rr
b
fR X
fG
fX X X R Xf
-
+ =
- +
(9)
k rs R G= (10)
( ) ( )
2 2
22 2
22
sm s
bk
s ss m ss rr ss s rr
b b
fX GV
fT
f fR G X X X X GR X
f f
=
+ - + +
(11)
( ) ( )
2 2
22 2
22
sm r s
be
s ss m ss rr ss s rr
b b
fX R sV
fT
f fR G X X X X GR X
f f
=
+ - + +
(12)
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Equations (11) and (12) are given in per unit, but the associated resistances are in ohms.
Figure 3. Mechanical and linear electrical torque versus slip.
Figure 4. Mechanical and electrical torque versus frequency curves per unit with Vsag= 10% .
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Figure 5. Mechanical and electrical torque versus frequency per unit with Vsag= 20%.
Figure 6. Mechanical and electrical torque versus frequency per unit with Vsag= 50%.
Figs. 3, 4, 5, and 6 illustrate that for lower wind speeds of 6 and 10 m/s, as the synchronous
frequencyfsand Vsagchange, the Teand Tmvalues of the rotor change in the same direction
as the frequency of the network, as shown in Tables III, IV, V, and VI. These figures and ta
bles give the results for V sag = 0% (i.e., only the frequency changes), 10%, 20%, and 50%.
However, for a higher wind speed of 13 m/s, asfsand Vsagchange, the Teand Tmvalues of
the rotor change in the opposite direction to the changes in the frequency of the network.
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For small changes in the slip according to the Kloss approach in equation (13), the torque
changes as follows [2]:
1 0m mT T Ka w= + D (14)
Then:
01 2 1
e
k mm
k
TT
s
w w
w
+ D= -
(15)
and
T TK
a
l
w l w
= =
(16)
or
0 0
4
0 , 00
1 1
2
p
MM
C
K R v Ta w l n rw l
= - (17)
Thus, the new angular operation speed[2] is
00 2 2
2
k k mm
k k e
k
k e
T TT
s s
Tk
sa
w
ww
w
- + -
D =
+(18)
w fs= 48 fs= 50 fs= 52
m pu Te pu m pu Te pu m pu Te pu
6 .96050 -.1157 1.0005 -.1064 1.0405 -.0974
10 .9621 -.5337 1.0021 -.491 1.0421 -.4493
13 .9631 -.7863 1.0035 -.8122 1.0439 -.8331
Table 3. Analytical MATLAB results for different frequencies.
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w fs= 48 fs= 50 fs= 52
m pu Te pu m pu Te pu m pu Te pu
6 .9606 -.1156 1.0006 -.1064 1.0406 -.0974
10 .9625 -.5163 1.0027 -.5137 1.0429 -.5086
13 .9738 -.7868 1.0043 -.8127 1.0448 -.8335
Table 4. Analytical MATLAB results for Vsag= 10%.
w fs= 48 fs= 50 fs= 52
m pu Te pu m pu Te pu m pu Te pu
6 .9607 -.1156 1.0007 -.1064 1.0407 -.0974
10 .9632 -.5163 1.0034 -.5136 1.0437 -.5085
13 .9648 -.7875 1.0054 -.8134 1.0461 -.8341
Table 5. Analytical MATLAB results for Vsag= 20%
w fs= 48 fs= 50 fs= 52
m pu Te pu m pu Te pu m pu Te pu
6 .9618 -.1153 1.0018 -.1061 1.0418 -.0971
10 .9681 -.5161 1.0088 -.5131 1.0494 -.5076
13 .9724 -.7927 1.0139 -.8181 1.0555 -.8382
Table 6. Analytical MATLAB results for Vsag= 50%
6. Simulation of wind generator with frequency change
During turbulence and changes in the grid frequency, the torque speed (slip) curves change
in such a way that as the frequency increases, the torque is increased at low wind speeds; 6
and 10 m/s, in contrast to Fig. 6 and decreases at a high speed of 13 m/s [2], as shown in
Table 7 and Figs. 715.
w fs= 48 fs= 50 fs= 52
m pu Te pu m pu Te pu m pu Te pu
6 .9619 -.1148 1.0019 -.1057 1.0418 -.0969
10 .9684 -.5179 1.0091 -.5134 1.0494 -.5076
13 .9724 -.7945 1.0147 -.8177 1.0559 -.8373
Table 7. Simulink simulation results for one- and two-mass shaft models
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Figs. 715 show the electrical torque and mechanical speed of the induction machine for
the one- and two-mass shaft turbine models at wind speeds of 6, 10, and 13 m/s to vali
date Table 7.
Figure 7. Electrical torque when = 48 and = 6m/s.
Figure 8. Electrical torque when fs= 50 and w= 6m/s.
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Figure 9. Electrical torque when fs= 52 and w= 6m/s.
Figure 10. Electrical torque when fs= 48 and w= 10m/s.
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Figure 11. Electrical torque when fs= 50 and w= 10m/s.
Figure 12. Electrical torque when fs= 52 and w= 10m/s.
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Figure 13. Electrical torque when fs= 48 and w= 13m/s.
Figure 14. Electrical torque when fs= 50 and w= 13m/s.
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Figure 15. Electrical torque when fs= 52 and w= 13m/s.
7. Simulation of wind station with one-mass and two-mass shaft turbine
models
The results of simulations of a simple grid, fixed-speed induction machine, and one-mass
and two-mass shaft turbines are given in Tables 8 -10 and Figs. 1642. For an induction wind
generator using the induction block in SIMULINK with high voltage sag i.e. 50% with fre
quencies 50 and 52 and equal to 13, Cpbecomes negative, and the results are unrealistic.
Then results of 50% voltage sag are realistic in new simulation of induction machine in Ta
bles 8 -10.
w fs= 48 fs= 50 fs= 52
m pu Te pu m pu Te pu m pu Te pu
6 .9624 -.1152 1.0024 -.106 1.0423 -.097
10 .9703 -.516 1.0111 -.5128 1.0519 -.5071
13 .9757 -.795 1.0176 -.8201 1.0595 -.8399
Table 8. Simulation results by SIMULINK for one and two mass shaft model for Vsag= 10%
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w fs= 48 fs= 50 fs= 52
m pu Te pu m pu Te pu m pu Te pu
6 .963 -.1151 1.003 -.1059 1.043 -.0969
10 .973 -.5159 1.014 -.5125 1.055 -.5066
13 .9799 -.7977 1.0223 -.8226 1.0648 -.842
Table 9. Simulation results by SIMULINK for one and two mass shaft model for Vsag= 20%
w fs= 48 fs= 50 fs= 52
m pu Te pu m pu Te pu m pu Te pu
6 .9674 -.114 1.0074 -.1048 1.0474 -.0959
10 .9933 -.5146 1.0364 -.5096 1.0796 -.502
13 1.0248 -.8239 1.0474 -.8347 1.0917 -.85
Table 10. Simulation results by SIMULINK for one and two mass shaft model for Vsag= 50%
Figure 16. Torque-time in per unit while Vsag= 10% and w= 6m/s, fs=48
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Figure 17. Torque-time in per unit while Vsag= 10% and w= 10m/s, fs= 48
Figure 18. Torque-time in per unit while Vsag= 10% and w= 13m/s, fs= 48
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Figure 19. Torque-time in per unit while Vsag= 20% and w= 6m/s, fs= 48
Figure 20. Torque-time in per unit while Vsag= 20% and w= 10m/s, fs= 48
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Figure 21. Torque-time in per unit while Vsag= 20% and w= 13m/s, fs= 48
Figure 22. Torque-time in per unit while Vsag= 50% and w= 6m/s, fs= 48
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Figure 23. Torque-time in per unit while Vsag= 50% and w= 10m/s, fs= 48
Figure 24. Torque-time in per unit while Vsag= 50% and w= 13m/s, fs= 48
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Figure 25. Torque-time in per unit while Vsag= 10% and w= 6m/s, fs= 50
Figure 26. Torque-time in per unit while Vsag= 10% and w= 10m/s, fs= 50
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Figure 27. Torque-time in per unit while Vsag= 10% and w= 13m/s, fs= 50
Figure 28. Torque-time in per unit while Vsag= 20% and w= 6m/s, fs= 50
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Figure 31. Torque-time in per unit while Vsag= 50% and w= 6m/s, fs= 50
Figure 32. Torque-time in per unit while Vsag= 50% and w= 10m/s, fs= 50
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Figure 33. Torque-time in per unit while Vsag= 50% and w= 13m/s, fs= 50 in new simulation of wind generator
Figure 34. Torque-time in per unit while Vsag=10% and w= 6m/s, fs= 52
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Figure 35. Torque-time in per unit while Vsag=10% and w= 10m/s, fs= 52
Figure 36. Torque-time in per unit while Vsag= 10% and w= 13m/s, fs= 52
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Figure 37. Torque-time in per unit while Vsag= 20% and w= 6m/s, = 52
Figure 38. Torque-time in per unit while Vsag= 20% and w= 10m/s, fs= 52
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Figure 39. Torque-time in per unit while Vsag= 20% and w= 13m/s, fs= 52
Figure 40. Torque-time in per unit while Vsag= 50% and w= 6m/s, fs= 52
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Figure 41. Torque-time in per unit while Vsag= 50% and w= 10m/s, fs= 52
Figure 42. Torque-time in per unit while Vsag= 50% and w= 13m/s, fs= 52 in new simulation of wind generator
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8. New Simulation of Induction Machine
Figs. 33 and 42 show the results of new simulation of the induction machine model illustrat
ed in Fig. 43 [1]. The new simulation, which has no limiters and switches, is used becauseat high grid voltage drop-down or sag, the Simulink induction model does not yield realistic results.
Figure 43. Induction machine Model in dqosystem
The new simulation of induction machine is in dqosystem and synchronous reference frame
simulation on the stator side; n (Transfer coefficient) is assumed to be 1. Circuit theory is
used in this simulation, and it does not have saturation and switch blocks, unlike the MAT
LABSIMULINK Induction block. In Fig. 43, L M is the magnetic mutual inductance, and r
andL are the ohm resistance and self-inductance of the dqocircuits, respectively. The ma
chine torque is given by equation (19). In this equation, id,qsandd,qs, the current and flux pa
rameters, respectively, are derived from linear equations (20)(23); they are sinusoidal
because the voltage sources are sinusoidal.
( )3
2 2e ds qs qs ds
PT i il l
= -
(19)
Where P is poles number, dsand qsare flux linkages and leakages, respectively, and iqsand
idsare stator currents in qand d circuits of dqosystem, respectively.
Then imatrix produced by the matrix is given by equation (20).
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( ) ( )
( ) ( ) ( )
1 1
1 1.
qdos qdoss s s s sr r
Tqdor qdor
r sr s r r r
iK L K K L K
iK L K K L K
l
l
- -
- -
=
(20)
where the inductance matrix parameters are given by (21), (22), (23).
( )1
0 0
0 0
0 0
ls M
s s s ls M
ls
L L
K L K L L
L
-
+ = +
(21)
( )1
0 00 0
0 0
lr M
r r r lr M
lr
L LK L K L L
L
- + = +
(22)
( ) ( ) ( )1 1
0 0
0 0
0 0 0
MT
s sr r r sr s M
L
K L K K L K L- -
= =
(23)
The linkage and leakage fluxes are given by (24) to (29).
( )qs ls qs M qs qr L i L i il = + + (24)
( )ds ls ds M ds dr L i L i il = + + (25)
os ls osL il = (26)
( )qr lr qr M qs qr L i L i il = + + (27)
( )dr lr dr M ds dr L i L i il = + + (28)
or lr or L il = (29)
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To create the torque in equation (19), it is necessary to determine the currents in equations(30)(33) from the stator and rotor currents by using current meters.
qsqs s qs ds
dr i
dt
ln wl= + + (30)
dsds s ds qs
dr i
dt
ln wl= - + (31)
( ) qrqr r qr r dr d
r idt
ln w w l
= + - + (32)
( ) drdr r dr r qr d
r idt
ln w w l = - - + (33)
9. Conclusion
As frequency changes and voltage sag occurs because of turbulence in wind stations in ride-through faults, the systems set point changes. The theoretical and simulation results resultsare similar for one mass shaft and two mass shaft turbine models. At lower wind speeds; 6and 10 m/s, the directions of the changes in the new working point are the same as those ofthe frequency changes. At a higher wind speed; 13 m/s, the directions of these changes areopposite to the direction of the frequency changes. Simulation results of high grid voltagesag with SIMULINK induction block has error and new simulation of wind induction generator in synchronous reference frame is presented without error and in 50% voltage sag, newsimulation of wind generator model has higher precision than that in 10% and 20% voltagesags; however, this model can simulate wind generator turbulence with voltage sags higherthan 50%. Although results of new simulation of induction machine with wind turbine for
50% voltage sag and frequencies 50 and 52 have been presented in this chapter.
10. Nomenclature
P =Generator power
= Air density
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A= Turbine rotor area
Cp =Power Coefficient
w = Wind speed
pitch =Pitch angle
Te = Electrical torque
Tm =Mechanical torque
J = Inertia
m = Mechanical speed
C = Drag coefficient
=Gear box ration
R = Blade radius
Rs= Stator resistance
L s= Stator inductance
L m= Mutual inductance
L r = Rotor inductance
R r = Rotor resistance
p =Pole pairs
=Stiffness
r,s = Rotor and stator flux
Kr,s =Rotor and stator park transformation in synchronous reference frame
ir,s =Rotor and stator current
vr,s =Rotor and stator voltage
11. Future Work
The new simulation of induction generator will be tested by new innovative rain turbine
theory and model of the author.
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Acknowledgements
I appreciate Dr. Oriol Gomis Bellmunt for conceptualization, Discussions and new informa
tion and Dr. Andreas Sumper for discussions about first part of chapter, with special thanks
to Dr. Joaquin Pedra for checking reference frame and starting point in new simulation of
induction machine.
Author details
Hengameh Kojooyan Jafari*
Address all correspondence to: [email protected]
Department of Electrical Engineering, Islamic Azad University, Islamshahr Branch, Iran
References
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[2] Sunmper, A., Gomis-Bellmunt, O., Sudria-Andreu, A., et al. (2009). Response of Fixed
Speed Wind Turbines to System Frequency Disturbances. ICEE Transaction on Power
Systems, 24(1), 181-192.
[3] Junyent-Ferre, A., Gomis-Bellmunt, O., Sunmper, A., et al. (2010). Modeling and con
trol of the doubly fed induction generator wind turbine. Simulation modeling practice
and theory journal of ELSEVIER, 1365-1381.
[4] Lubosny, Z. (2003). Wind Turbine operation in electric power systems. Springer pub
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